Error analysis of the Legendre-Gauss collocation methods for the nonlinear distributed-order fractional differential equation

Atanackovic, 2002, A generalized model for the uniaxial isothermal deformation of a viscoelastic body, Acta Mech., 159, 77, 10.1007/BF01171449 Atanacković, 2007, On a nonlinear distributed order fractional differential equation, J. Math. Anal. Appl., 328, 590, 10.1016/j.jmaa.2006.05.038 Bhrawy, 2015, A method based on the Jacobi tau approximation for solving multi-term time-space fractional partial differential equations, J. Comput. Phys., 281, 876, 10.1016/j.jcp.2014.10.060 Brunner, 2004 Caputo, 1967, Linear models of dissipation whose Q is almost frequency independent–II, Geophys. J. R. Astron. Soc., 13, 529, 10.1111/j.1365-246X.1967.tb02303.x Caputo, 2001, Distributed order differential equations modelling dielectric induction and diffusion, Fract. Calc. Appl. Anal., 4, 421 Chen, 2016, Finite difference/spectral approximations for the distributed order time fractional reaction–diffusion equation on an unbounded domain, J. Comput. Phys., 315, 84, 10.1016/j.jcp.2016.03.044 Chen, 2010, Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equations with a weakly singular kernel, Math. Comput., 79, 147, 10.1090/S0025-5718-09-02269-8 Dehghan, 2017, Spectral element technique for nonlinear fractional evolution equation, stability and convergence analysis, Appl. Numer. Math., 119, 51, 10.1016/j.apnum.2017.03.009 Dehghan, 2018, An efficient technique based on finite difference/finite element method for solution of two-dimensional space/multi-time fractional Bloch-Torrey equations, Appl. Numer. Math., 131, 190, 10.1016/j.apnum.2018.04.009 Dehghan, 2016, Legendre spectral element method for solving time fractional modified anomalous sub-diffusion equation, Appl. Math. Model., 40, 3635, 10.1016/j.apm.2015.10.036 Dehghan, 2010, Solving nonlinear fractional partial differential equations using the homotopy analysis method, Numer. Methods Partial Differ. Equ., 26, 448, 10.1002/num.20460 Dehghan, 2016, The dual reciprocity boundary integral equation technique to solve a class of the linear and nonlinear fractional partial differential equations, Math. Methods Appl. Sci., 39, 2461, 10.1002/mma.3707 Diethelm, 2009, Numerical analysis for distributed-order differential equations, J. Comput. Appl. Math., 225, 96, 10.1016/j.cam.2008.07.018 Doha, 2011, Efficient Chebyshev spectral methods for solving multi-term fractional orders differential equations, Appl. Math. Model., 35, 5662, 10.1016/j.apm.2011.05.011 Doha, 2012, A new Jacobi operational matrix: an application for solving fractional differential equations, Appl. Math. Model., 36, 4931, 10.1016/j.apm.2011.12.031 Donchev, 1998, Stability and Euler approximation of one-sided Lipschitz differential inclusions, SIAM J. Control Optim., 36, 780, 10.1137/S0363012995293694 Ervin, 2007, Variational solution of fractional advection dispersion equations on bounded domains in Rd, Numer. Methods Partial Differ. Equ., 23, 256, 10.1002/num.20169 Gao, 2015, Some high-order difference schemes for the distributed-order differential equations, J. Comput. Phys., 298, 337, 10.1016/j.jcp.2015.05.047 Gao, 2015, Two alternating direction implicit difference schemes with the extrapolation method for the two-dimensional distributed-order differential equations, Comput. Math. Appl., 69, 926, 10.1016/j.camwa.2015.02.023 Guo, 2004, Jacobi approximations in non-uniformly Jacobi-weighted Sobolev spaces, J. Approx. Theory, 128, 1, 10.1016/j.jat.2004.03.008 Guo, 2009, Legendre–Gauss collocation methods for ordinary differential equations, Adv. Comput. Math., 30, 249, 10.1007/s10444-008-9067-6 Jiao, 2012 Katsikadelis, 2014, Numerical solution of distributed order fractional differential equations, J. Comput. Phys., 259, 11, 10.1016/j.jcp.2013.11.013 Kharazmi, 2017, Petrov-Galerkin and spectral collocation methods for distributed order differential equations, SIAM J. Sci. Comput., 39, A1003, 10.1137/16M1073121 Li, 1983 Li, 2009, A space-time spectral method for the time fractional diffusion equation, SIAM J. Numer. Anal., 47, 2108, 10.1137/080718942 López-Fernández, 2004, On the numerical inversion of the Laplace transform of certain holomorphic mappings, Appl. Numer. Math., 51, 289, 10.1016/j.apnum.2004.06.015 López-Fernández, 2006, A spectral order method for inverting sectorial Laplace transforms, SIAM J. Numer. Anal., 44, 1332, 10.1137/050629653 Lorenzo, 2002, Variable order and distributed order fractional operators, Nonlinear Dyn., 29, 57, 10.1023/A:1016586905654 Lubich, 1983, Runge-Kutta theory for Volterra and Abel integral equations of the second kind, Math. Comput., 41, 87, 10.1090/S0025-5718-1983-0701626-6 Lubich, 1988, Convolution quadrature and discretized operational calculus. I, Numer. Math., 52, 129, 10.1007/BF01398686 Luchko, 2009, Boundary value problems for the generalized time-fractional diffusion equation of distributed order, Fract. Calc. Appl. Anal., 12, 409 Luchko, 2010, Some uniqueness and existence results for the initial-boundary-value problems for the generalized time-fractional diffusion equation, Comput. Math. Appl., 59, 1766, 10.1016/j.camwa.2009.08.015 Morgado, 2015, Numerical approximation of distributed order reaction–diffusion equations, J. Comput. Appl. Math., 275, 216, 10.1016/j.cam.2014.07.029 Morgado, 2017, Numerical solution for diffusion equations with distributed order in time using a Chebyshev collocation method, Appl. Numer. Math., 114, 108, 10.1016/j.apnum.2016.11.001 Podlubny, 1999 Qazi, 2006, The mean value theorem and analytic functions of a complex variable, J. Math. Anal. Appl., 324, 30, 10.1016/j.jmaa.2005.11.049 Saadatmandi, 2010, A new operational matrix for solving fractional-order differential equations, Comput. Math. Appl., 59, 1326, 10.1016/j.camwa.2009.07.006 Schädle, 2006, Fast and oblivious convolution quadrature, SIAM J. Sci. Comput., 28, 421, 10.1137/050623139 Shen, 2011 Weilbeer, 2005 Ye, 2015, Numerical analysis for the time distributed-order and Riesz space fractional diffusions on bounded domains, IMA J. Appl. Math., 80, 825, 10.1093/imamat/hxu015 Zhang, 2018, A Crank-Nicolson ADI Galerkin-Legendre spectral method for the two-dimensional Riesz space distributed-order advection-diffusion equation, Comput. Math. Appl., 76, 2460, 10.1016/j.camwa.2018.08.042